The Annals of Probability

Stochastic equations, flows and measure-valued processes

Donald A. Dawson and Zenghu Li

Full-text: Open access

Abstract

We first prove some general results on pathwise uniqueness, comparison property and existence of nonnegative strong solutions of stochastic equations driven by white noises and Poisson random measures. The results are then used to prove the strong existence of two classes of stochastic flows associated with coalescents with multiple collisions, that is, generalized Fleming–Viot flows and flows of continuous-state branching processes with immigration. One of them unifies the different treatments of three kinds of flows in Bertoin and Le Gall [Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 307–333]. Two scaling limit theorems for the generalized Fleming–Viot flows are proved, which lead to sub-critical branching immigration superprocesses. From those theorems we derive easily a generalization of the limit theorem for finite point motions of the flows in Bertoin and Le Gall [Illinois J. Math. 50 (2006) 147–181].

Article information

Source
Ann. Probab., Volume 40, Number 2 (2012), 813-857.

Dates
First available in Project Euclid: 26 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1332772722

Digital Object Identifier
doi:10.1214/10-AOP629

Mathematical Reviews number (MathSciNet)
MR2952093

Zentralblatt MATH identifier
1254.60088

Subjects
Primary: 60G09: Exchangeability 60J68: Superprocesses
Secondary: 60J25: Continuous-time Markov processes on general state spaces 92D25: Population dynamics (general)

Keywords
Stochastic equation strong solution stochastic flow coalescent generalized Fleming–Viot process continuous-state branching process immigration superprocess

Citation

Dawson, Donald A.; Li, Zenghu. Stochastic equations, flows and measure-valued processes. Ann. Probab. 40 (2012), no. 2, 813--857. doi:10.1214/10-AOP629. https://projecteuclid.org/euclid.aop/1332772722


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